Experimental Evidence for Longitudinal Scaling Exponent Saturation in Shear Turbulence

The asymptotic behavior of velocity statistics in the tails of distributions and at high Reynolds numbers remains unresolved in turbulence. To investigate this behavior we measured the $n$th-order moments of the distributions of longitudinal velocity differences, $S_n(r) \equiv \langle [u(x+r)-u(x)]^n \rangle \sim r^{\zeta_n}$, in turbulent shear layers at Taylor-scale Reynolds numbers up to $Re_\lambda \approx 1400$. We used a nanoscale hot-wire probe with a sensing length, $l_w$, that was about half the Kolmogorov scale, $\eta$. We obtained datasets that were up to $5\times 10^7$ integral timescales long, so that the statistics converged up to $n=14$. In the inertial range, the exponents, $\zeta_n$, deviate from classical models and appear to saturate near $\zeta_n \approx 2.2 \pm 0.1$ for $n \gtrsim 12$. The saturation in the exponents is supported by a collapse of the tails of the velocity-difference distributions, and by plateaus in their compensated moments. These results constitute the first experimental evidence for scaling exponent saturation in longitudinal velocity increments, and is consistent with a dominance of localized vortex filaments in turbulence.